Interactions of Two-Dimensionally Confined Electrons with an Adjacent Magnetic Monopole

Abstract

An electron in the presence of a magnetic monopole cannot form a bound state to the monopole in three dimensions. All states formed are scattering and follow a geodesic trajectory on the surface of a cone. In this thesis I show that confining the electron to two dimensions and placing the monopole above or below said plane allows for bound states to be formed. Classically, utilising Lagrangian mechanics, these are fully bound never forming scattering states without an influx of energy. Quantum mechanically (Solutions to Schrödinger time independent equation) and semi-classically (Bohr- Sommerfeld quantisation, WKB approximation), these states are quasi-bound with finite lifetimes before turning into a scattering state. The minimum charge that can bind an electron to a magnetic monopole is approximately the same strength as 16 Dirac monopoles. The lifetimes of these scattering states is dependent on the electron's energy eigenvalue, the strength of the magnetic monopole, and the distance the monopole is from the plane. Magnetic monopoles can be detected using a SQUID (Superconducting QUantum Interference Device) measuring the quantised jumps in magnetic flux. In this thesis I ask: can they be detected using the Hall effect? With the electrons bound to a plane permeated by the non-uniform magnetic field produced by a magnetic monopole; what will the Hall voltage look like across the plane, and can it be measured? Magnetohydrodynamics is utilised to model the flow of an electron gas by treating it as a fluid that interacts with both magnetic and electrical fields. A single Dirac monopole produces a peak in Hall voltage across the modelled Hall sensor of the order 10−6V. For a monopole found in spin-ice, which is about 1/8000 the magnetic charge of a single Dirac monopole, this voltage is considerably less